Extension of a theorem of Kiming and Olsson for the partition function (Q5943745)

Let \(p(n)\) denote the number of unrestricted partitions of the nonnegative integer \(n\). \textit{I. Kiming} and \textit{J. B. Olsson} [Arch. Math. 59, 348-360 (1992; Zbl 0765.11041)] proved that if \(q>3\) is prime and \(r\) is an integer for which \(p(qn+r)\equiv 0\pmod q\) for all \(n\), then \(24r\equiv 1\pmod q\). In the paper under review the author examines the rarity of these Ramanujan-type congruences. The main result of the paper is the following Theorem 1. Let \(q>3\) be prime and \(m>1\) be an odd, square-free integer. If for every \(r\pmod q\) with \(24r\not\equiv 1\pmod q\) there exists an \(n_r\) such that \(n_r\equiv r\pmod q\) and \(p(n_r)\not\equiv 0\pmod m\), then for every \(k\geq 1\) there are infinitely many \(n\) for which \(p(q^kn+ \delta)\not\equiv 0\pmod m\) provided \(24\delta\not\equiv 1\pmod q\). The title of the paper refers to the following Corollary 2. If \(q>3\) is prime and \(\gamma\) is an integer for which \(24\gamma\not\equiv 1\pmod q\) then for every positive integer \(k\) there are infinitely many integers \(n\) for which \(p(q^kn+ \gamma)\not\equiv 0\pmod q\). The proof uses the results of \textit{K. Ono} [Math. Ann. 312, 251-260 (1998; Zbl 0914.11054)] and the author [Int. J. Math. Math. Sci. 23, 493-496 (2000; Zbl 0993.11054)].